Stability of discrete-time systems in the simplest critical case

The paper considers nonlinear time-invariant discrete-time systems of arbitrary orders. The right-hand sides of the equations are sums of polynomials of arbitrary degrees and continuous functions which are infinitely small with respect to the highest-degree terms of the polynomials. It is assumed that the characteristic polynomial of the linearized system has one root equal to unity, while its other roots lie inside the unit disk.